A 2D kernel determination problem in a visco-elastic porous medium with a weakly horizontally inhomogeneity

We consider a system of hyperbolic integro-differential equations of SH waves in a visco-elastic porous medium. In this work, it is assumed that the visco-elastic porous medium has weakly horizontally inhomogeneity. The direct problem is the initial-boundary problem: the initial data is equal to zero, and the Neumann-type boundary condition is specified at the half-plane boundary and is an impulse function. As additional information, the oscillation mode of the half-plane line is given. It is assumed that the unknown kernel has the form K(x,t)=K₀(t)+ϵxK₁(t)+…, where ϵ is a small parameter. In this work, we construct a method for finding K₀,K₁ up to a correction of the order of O(ϵ²).